We calculate the cross section and scattered-muon distributions for the process ..mu..+Z..--> mu..+..nu..+nu-bar+Z, namely neutrino pair production by muons in the Coulomb field of a heavy nucleus. The cross section is a measure of the number of light-neutrino flavors, assuming the standard model, as previously recognized. The differential distributions of the scattered muon are needed to interpret any partial measurement (short of a total cross section) and also provide important consistency checks for an eventual experimental signal. Unfortunately, the predicted cross section is very small for a small number of flavors; also, the final muon energy peaks near zero so that the practical acceptance is reduced. A measurement of this cross section is therefore not foreseen in the near future, despite its interest. We also calculate the contribution expected from diffractive vector-meson production ..mu..+Z..--> mu..+V/sup 0/+Z with V/sup 0/..--> nu..nu-bar decay; the V/sup 0/ = psi contribution considerably exceeds the direct neutrino rate. However, the V/sup 0/ process yields a completely different muon spectrum and can be distinguished by this; it also offers an independent measure of the number of neutrino flavors.